|
1
General guidelines for marking
|
0.00 |
|
| 2 Formula for the power dissipation $P_{\mathrm{el}}=V^{2} / R_{j}$. | 0.50 |
|
| 3 Relating the power dissipation to the temperature of the resistor in oscillations-free stationary regime, $P_{\text {el }}=P=\alpha\left(T_{\text {eq }}-T_{0}\right)$ | 0.50 |
|
| 4 Expressing the voltage in terms of the temperature if the thermal equilibrium were to be reached, $V=$ $\sqrt{R_{j} \alpha\left(T_{\text {eq }}-T_{0}\right)}$. | 0.50 |
|
| 5 $V$ is not expressed explicitly. | -0.10 |
|
| 6 Realising that oscillations will not happen if $V>$ $\sqrt{R_{2} \alpha\left(T_{\text {eq }}-T_{0}\right)}$ or $V<\sqrt{R_{1} \alpha\left(T_{\text {eq }}-T_{0}\right)}$. No marks if only one inequality is obtained (but no subtractions because of that in a3 - in most cases those who got correct expression for one of the voltages but has a wrong or missing expression for the other gets full marks for a1-a3, and 0 pts for a4). | 0.50 |
|
|
1
Realising that the $I-t$ curve is made of segments of exponents, joined without discontinuities. Full marks can be given if there is no $I-t$ graph, but the $T-t$ graph is made of the segments of vanishing exponents, connected with temperature jumps in a correct direction |
1.00 |
|
|
2
Partial credit of 0.5 pts if it is made of curved segments for which it is not clear that these are exponents, or if these are growing exponents, but which are connected continuously with a discontinuous derivative $\frac{\mathrm{d} I}{\mathrm{~d} t}$. Partial credit of 0.5 pts if the segments of the $T-t$ are either growing exponents or curves of unclear shape, still connected so that it would correspond to a continuous $I(t)$-curve with a discontinuous derivative. Partial credit of 0.5 pts is given if there is no $I-t$-curve shown, but $V-t$ curve is shown to be made of decaying exponential segments, connected with jumps. |
0.50 |
|
| 3 No points if $I(t)$ is discontinuous, or if only one segment of an exponent is shown. | 0.00 |
|
| 4 Realising that (i) one of these exponents is in a form $a_{1}-b_{1} \mathrm{e}^{-t / \tau_{1}}$ and (ii) the other one - in a form $a_{2}+b_{2} \mathrm{e}^{-t / \tau_{2}}$ where (iii) the $a_{1}>a_{2}$ and (iv) $\tau_{1}>\tau_{2}$. It is not necessary to write down these inequalities mathematically - it is enough it these are clear from a sketch. Inequality $\tau_{1}>\tau_{2}$ does not need to be written if expressions for $\tau_{1}$ and $\tau_{2}$ are given. Full marks can be given if $I-t$ graph is missing, but $T-t$ graph is correct and has all the features as described in b6. Full marks can be also given if the correct exponential forms are documented not here, but in part c. | 0.00 |
|
| 5 $a_{1}-b_{1} \mathrm{e}^{-t / \tau_{1}}$ | 0.30 |
|
| 6 $a_{2}+b_{2} \mathrm{e}^{-t / \tau_{2}}$ | 0.30 |
|
| 7 $a_{1}>a_{2}$ | 0.30 |
|
| 8 $\tau_{1}>\tau_{2}$ | 0.10 |
|
| 9 Realising that this exponential behaviour breaks down once the critical temperature is reached. This does not need to be written specifically if the jumps in $T-t$ graph happen at $T=T_{c}$. No marks are given if there is no clear discontinuity of $T$ at $T_{c}$ and/or if there are discontinuities of $T(t)$ or $\frac{\mathrm{d} T}{\mathrm{~d} t}$ at some other values of $T$. | 1.00 |
|
| 10 Relating the critical temperature to the corresponding critical current $I_{j}$ | 0.50 |
|
| 11 Realising that the temperature curve $T(t)$ is related to $I(t)$-curve, $T(t)=T_{0}+\frac{R(t) I(t)^{2}}{\alpha}$ | 0.50 |
|
| 12 Drawing a correct final sketch which has the following features: exponential segments showing an exponential relaxation of $T(t)$ in a right direction both when $R=R_{1}$ and when $R=R_{2}$; jumps in a right direction each time when $T$ reaches $T_{c}$. No points are given if any of the listed features is missing. | 1.00 |
|
| 13 Subtract $0.2$ for each missing label on the axes and also if the temperature jumps do not occur at the same value of $T$. | 5 × -0.20 |
|
| 14 Using the feature from the graph that the maximal and minimal temperatures are taken immediately after a phase transition when $I=I_{1}$ and $I=I_{2}$ | 0.50 |
|
| 15 Correct answer for the ratio of the maximal and minimal temperatures. | 0.50 |
|
| 16 Only $0.3$ pts if the answer is not simplified. | 0.30 |
|
| 1 (1) Expressing the duration of each of the exponential segments as $t_{j}=\frac{L}{R_{j}} \ln \frac{\Delta I_{j, i}}{\Delta I_{j, f}}$ where $\Delta I_{j, i}$ and $\Delta I_{j, f}$ denote the corresponding initial and final departures of the current from the equilibrium value (full marks to be given if the final answer is correct). | 0.50 |
|
| 2 $60 \%$ of points if $t_{j}$ is related to $\Delta I_{j, i}$ and $\Delta I_{j, f}$ correctly, but not expressed explicitly. | 0.30 |
|
| 3 (2) Expressing the duration of each of the exponential segments as $t_{j}=\frac{L}{R_{j}} \ln \frac{\Delta I_{j, i}}{\Delta I_{j, f}}$ where $\Delta I_{j, i}$ and $\Delta I_{j, f}$ denote the corresponding initial and final departures of the current from the equilibrium value (full marks to be given if the final answer is correct). | 0.50 |
|
| 4 $60 \%$ of points if $t_{j}$ is related to $\Delta I_{j, i}$ and $\Delta I_{j, f}$ correctly, but not expressed explicitly. | 0.30 |
|
| 5 Subtract $0.2$ for each incorrect $\Delta I_{j, i}$ and $\Delta I_{j, f}$, $i=1,2$ (this means that if none of them is correct, only $0.2$ pts are given for c1). | 5 × -0.20 |
|
| 6 Correct first terms in the final answer | 0.50 |
|
| 7 $40\%$ of it if the answer is not simplified | 0.20 |
|
| 8 Correct second terms in the final answer | 0.50 |
|
| 9 $40\%$ of it if the answer is not simplified | 0.20 |
|